Question

△yz is equilateral. complete the proof that △xyz≌△xwv.

1 △yz is equilateral given
2 ∠w≌∠y given
3 ∠yxz≌∠vxw
4 xz≌vx
5 △xyz≌△xwv aas

Explanation:

Step1: Vertical angles are congruent

Vertical angles are formed when two lines intersect. In the given figure, \(\angle YXZ\) and \(\angle VWX\) are vertical angles. By the vertical - angles theorem, if two angles are vertical angles, then they are congruent. So, \(\angle YXZ\cong\angle VWX\) because vertical angles are congruent.

Step2: Properties of an equilateral triangle

Since \(\triangle YVZ\) is equilateral, all its sides are equal. In particular, \(YZ = VZ\). But in the context of the triangles \(\triangle YXZ\) and \(\triangle VWX\), we know that in an equilateral triangle \(\triangle YVZ\), \(YZ = VZ\). Also, from the AAS (Angle - Angle - Side) congruence criterion, we need a side. Since \(\triangle YVZ\) is equilateral, \(YZ=VZ\). But looking at the triangles \(\triangle YXZ\) and \(\triangle VWX\), we note that in an equilateral triangle all sides are equal. So, \(YZ = VZ\) (sides of an equilateral triangle are equal).

Answer:

1. Vertical angles are congruent; 4. \(YZ = VZ\) (sides of an equilateral triangle are equal)